A304785 -id:A304785 - cp's OEIS frontend

A304786 Expansion of Product_{k>=1} (1 - q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

1, -1, -1, -1, 0, 1, -1, 4, 2, 3, 1, 8, -8, 10, -8, -9, -15, -6, -46, -14, -65, -28, 14, -29, -43, -37, 298, 59, 234, 165, 738, 354, 1083, 703, 1944, -2024, 1917, -1085, 3658, -2385, -6421, -7220, 118, -15569, -11604, -19162, -9448, -36140, -24561, -50505, -24807, 47645

Offset: 0

Ilya Gutkovskiy, May 18 2018

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Convolution inverse of A270995.

Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions

Cf. A000009, A270995, A271619, A304783, A304785.

nmax = 51; CoefficientList[Series[Product[(1 - PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 51}]

G.f.: Product_{k>=1} (1 - A000009(k)*x^k).

A316230 Expansion of Product_{k>=1} 1/(1 + p(k)*x^k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -1, -2, 1, -2, 2, 0, 24, -17, 31, -21, 94, -107, 121, -443, 742, -977, 532, -2159, 3275, -6193, 6988, -11156, 30278, -39214, 42759, -80255, 149070, -193093, 291229, -451125, 1017812, -1335848, 1609412, -3248202, 5606551, -7684574, 10012531, -17908468

Offset: 0

Ilya Gutkovskiy, Jun 27 2018

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Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

Cf. A000041, A063834, A271619, A304784, A304785, A316231.

nmax = 39; CoefficientList[Series[Product[1/(1 + PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsP[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-PartitionsP[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 39}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*p(j)^k*x^(j*k)/k).

cp's OEIS Frontend

A304786 Expansion of Product_{k>=1} (1 - q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

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